$$$- 3 \cos{\left(x \right)}$$$の積分
入力内容
$$$\int \left(- 3 \cos{\left(x \right)}\right)\, dx$$$ を求めよ。
解答
定数倍の法則 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ を、$$$c=-3$$$ と $$$f{\left(x \right)} = \cos{\left(x \right)}$$$ に対して適用する:
$${\color{red}{\int{\left(- 3 \cos{\left(x \right)}\right)d x}}} = {\color{red}{\left(- 3 \int{\cos{\left(x \right)} d x}\right)}}$$
余弦の積分は$$$\int{\cos{\left(x \right)} d x} = \sin{\left(x \right)}$$$:
$$- 3 {\color{red}{\int{\cos{\left(x \right)} d x}}} = - 3 {\color{red}{\sin{\left(x \right)}}}$$
したがって、
$$\int{\left(- 3 \cos{\left(x \right)}\right)d x} = - 3 \sin{\left(x \right)}$$
積分定数を加える:
$$\int{\left(- 3 \cos{\left(x \right)}\right)d x} = - 3 \sin{\left(x \right)}+C$$
解答
$$$\int \left(- 3 \cos{\left(x \right)}\right)\, dx = - 3 \sin{\left(x \right)} + C$$$A